Complete flags on flat vector bundles in positive characteristic

Let $X$ be a connected, smooth, and projective curve of genus $g$ over an algebraically closed field of characteristic $p>0$. This paper investigates a characteristic-$p$ analogue of a well-known fact concerning flat vector bundles in characteristic $0$. That is to say, we prove that the inequality $g\le 1$ holds if and only if any flat vector bundle on $X$ admits a complete flag. We also explore a generalization of this result in a broader setting.